Nakanishi's conjecture for classical knots

Establish that every classical knot is 4-equivalent to the trivial knot, thereby resolving Nakanishi's conjecture.

Background

The paper recalls that Nakanishi conjectured in 1979 that every knot is 4-equivalent to the trivial knot. Although the conjecture has been verified for several classes, including closures of algebraic tangles, knots with crossing number at most 12, and closures of 3-strand braids, its general case remains unresolved.

The paper studies a welded analogue of 4-equivalence and proves results for closures of algebraic virtual tangles, but does not resolve the original conjecture for all classical knots.

References

In 1979, Nakanishi conjectured that every knot is 4-equivalent to the trivial knot. This is no longer true for links, the Hopf link being a classical counterexample. In 1985, Kawauchi proposed a link analogue of the conjecture, asserting that any two link-homotopic links are 4-equivalent. Both conjectures appear in. While Kawauchi's conjecture is now known to be false, Nakanishi's original conjecture remains open.

— A welded extension of 4-equivalence  (2609.04624 - Emmanuel, 4 Sep 2026) in Introduction